Abstract
Algebra with Applications provides a friendly and concise introduction to
algebra, with an emphasis on its uses in the modern world. The first part of
this book covers groups, after some preliminaries on sets, functions,
relations, and induction, and features applications such as public-key
cryptography, Sudoku, the finite Fourier transform, and symmetry in chemistry
and physics. The second part of this book covers rings and fields, and features
applications such as random number generators, error correcting codes, the
Google page rank algorithm, communication networks, and elliptic curve
cryptography. The book's masterful use of colorful figures and images helps
illustrate the applications and concepts in the text. Real-world examples and
exercises will help students contextualize the information. Meant for a
year-long undergraduate course in algebra for mathematics, engineering, and
computer science majors, the only prerequisites are calculus and a bit of
courage when asked to do a short proof.

This book gives an introduction to
harmonic analysis on symmetric spaces, focusing on advanced topics such as higher
rank spaces, positive definite matrix space and generalizations. It is intended
for beginning graduate students in mathematics or researchers in physics or
engineering. As with the earlier book entitled "Harmonic Analysis on
Symmetric Space—Euclidean Space, the Sphere, and
the Poincaré Upper Half Plane, the style is informal
with an emphasis on motivation, concrete examples, history, and applications.
The symmetric spaces considered here are quotients X=G/K, where G is a
non-compact real Lie group, such as the general linear group GL(n,R) of all n x n non-singular real matrices, and K=O(n),
the maximal compact subgroup of orthogonal matrices. Other examples are
Siegel's upper half "plane" and the quaternionic
upper half "plane". In the case of the general linear group, one can
identify X with the space Pn of n x n
positive definite symmetric matrices.

Many corrections and updates are
included in this new edition. Updates include discussions of random matrix theory
and quantum chaos, as well as recent research on modular forms and their
corresponding L-functions in higher rank.Many applications have been added, such as the solution of the heat
equation on Pn, the central limit theorem
of Donald St. P. Richards for Pn, results
on densest lattice packing of spheres in Euclidean space, and GL(n)-analogs of the Weyl law for eigenvalues of the
Laplacian in plane domains, as well as computations of analogues of Maass waveforms for GL(3).

Topics featured throughout the
text include inversion formulas for Fourier transforms, central limit theorems,
fundamental domains in X for discrete groups Γ (such as the modular group GL(n,Z) of n x n matrices with
integer entries and determinant ±1), connections with the problem of finding
densest lattice packings of spheres in Euclidean space, automorphic forms, Hecke operators, L-functions, and the Selberg
trace formula and its applications in spectral theory as well as number theory.

A movie related to this book showing
the projection of (t,v,x1,x2,x3) onto the
x-coordinates in the Grenier fundamental domain (see
page 151 of the old edition)for
GL(3,Z) acting on the determinant one surface in positive 3x3 matrix space as
the coordinates (t,v) travel along t=vfrom .9 to 1.03.

A movie related
to Volume I showing a big bang related to points in the fundamental domain of
the modular group Γ=SL(2,Z), which are Γ-equivalent to points on a
horocycle moving down toward the real axis.The y-axis has been distored so that infinity is at height 10.

Volume 1 gives an introduction to
harmonic analysis on the simplest symmetric spaces - Euclidean space, the
sphere, and the Poincaré upper half plane H and fundamental domains for
discrete groups of isometries such as SL(2,Z) in the
case of H. The emphasis is on examples, applications, history. The intention is
to be a friendly introduction for non-experts.

Volume 2 concerns higher rank
symmetric spaces and their fundamental domains for discrete groups of
isometries. Emphasis is on the general linear group G=GL(n,R) of invertible nxn real
matrices and its symmetric space G/K which we identify with the space Pn
of positive definite nxn real symmetric matrices.
Applications in multivariate statistics and the geometry of numbers are
considered.

Zeta
Functions of Graphs: A Stroll through the Garden,Cambridge U. Press, Cambridge, U.K.,
2011.Available
as an ebook.

Graph theory meets number theory
in this book. Ihara zeta functions of finite graphs are reciprocals of
polynomials, sometimes in several variables. Analogies abound with number-theoretic
functions such as Riemann or Dedekind zeta functions. For example, there is a
Riemann hypothesis (which may be false) and a prime number theorem for graphs.
Explicit constructions of graph coverings use Galois theory
to generalize Cayley and Schreier graphs. Then
non-isomorphic simple graphs with the same zeta function are produced, showing
you cannot “hear” the shape of a graph.The spectra of matrices such as the adjacency and edge adjacency
matrices of a graph are essential to the plot of this book, which makes
connections with quantum chaos and random matrix theory and also with expander
and Ramanujan graphs, of interest in computer science. Many well-chosen
illustrations and exercises, both theoretical and computer-based, are included
throughout.

This book gives a friendly introduction to Fourier analysis on finite
groups, both commutative and non-commutative. Aimed at students in mathematics,
engineering and the physical sciences, it examines the theory of finite groups
in a manner that is both accessible to the beginner and suitable for graduate
research. With applications in chemistry, error-correcting codes, data
analysis, graph theory, number theory and probability, the book presents a
concrete approach to abstract group theory through applied examples, pictures
and computer experiments.

In the first part, the author parallels
the development of Fourier analysis on the real line and the circle, and then
moves on to analogues of higher dimensional Euclidean space. The second part
emphasizes matrix groups such as the Heisenberg group of upper triangular 2x2 matrices.
The book concludes with an introduction to zeta functions on finite graphs via
the trace formula.

14) Joint with M. D. Horton and D.
Newland, The Contest between the Kernels in the Selberg Trace Formula for the
(q+1)-regular Tree, in Contemporary Mathematics, Volume 398 (2006), The
Ubiquitous Heat Kernel, Edited by Jay Jorgenson and Lynne Walling, pages
265-294.
http://www.ams.org/bookstore/conmseries

15) Joint with M. D. Horton and H.
M. Stark, What are Zeta Functions of Graphs and What are They Good For?, Contemporary Mathematics, Volume 415 (2006), Quantum
Graphs and Their Applications; Edited by Gregory Berkolaiko,
Robert Carlson, Stephen A. Fulling, and Peter Kuchment,
pages 173-190.

15) I was one of the many
lecturers in the ParkCity
summer research session which took place in Park City,
Utah from June 30 to July 20, 2002. The topic
was Automorphic Forms. I was there for the segment on quantum chaos. For
information about this program you can go to the website http://www.ias.edu/parkcity.

17) I organized a Special
Session on Zeta Functions of Graphs and Related Topics at the Fourth
International Conference on Dynamical Systems and Differential Equations to be
held May 24-27, 2002 in Wilmington, North Carolina. The aim of the session was
to discuss current work on the Ihara-Selberg functions attached to graphs and
related topics such as Ramanujan graphs, the trace formula on trees. The hope
was to emphasize connections between various fields such as graph theory,
topology, mathematical physics, number theory, dynamical systems. One example
is the connection between graph zeta functions and Jones polynomials of knots
found by Lin and Wang. The conference website is http://www.uncwil.edu/mathconf/.
Special session abstracts can be found at abstracts.htm.
Proceedings appeared in 2003 Supplement Volume of Discrete and
Continuous Dynamical Systems, devoted to the Proceedings of the Fourth
International Conference on Dynamical Systems and Differential Equations, May
24-27, 2002, at Wilmington, NC, Edited by W. Feng, S. Hu and X. Lu

The last one is a tessellation of the
finite upper half plane for the field with 11*11 elements coming from the group
of non-singular 2x2 matrices from the field with 11 elements.Explanations can be found in

AUDREY TERRAS received her B.S. degree in Mathematics from
the University of Maryland, College Park in 1964, where she was inspired by the
lectures of Sigekatu Kuroda to become a number
theorist. She was particularly impressed by the use of analysis (in particular
using zeta functions and multiple integrals) to derive algebraic results. She
received her M.A. (1966) and Ph.D. (1970) from Yale University.In 1972 she became an assistant professor of
mathematics at the University of California, San Diego. She became a full
professor at U.C.S.D. in 1983. She retired in 2010. She has had 25 Ph.D.
students. She is a fellow of the Association for Women in Mathematics, the
American Mathematical Society, and the American Association for the Advancement
of Science, has served on the Council of the American Mathematical Society,
gave the 2008 Noether lecture of the Association for
Women in Mathematics. She has published 5 books, helped to edit another, and
published lots of research papers. Her research interests include number
theory, harmonic analysis on symmetric spaces and finite groups (including
applications), special functions, algebraic graph theory, especially zeta
functions of graphs, arithmetical quantum chaos, and Selberg’s
trace formula. When
lecturing on mathematics, she believes it is important to give examples,
applications and colorful pictures.